It is not enough to say, over and over again, that Greece invented democracy. In reality, the ancient Greeks invented politics itself. They were the first people to think about the state as an intellectual problem. Their attempts to reason about their complex ways of life gave us a huge range of political terms that still define politics today – and all these words described realities in the ancient Greek world. So if modern Greeks have a heritage of democracy, they can also lay claim to tyranny, oligarchy, aristocracy, plutocracy, and monarchy – Greek words all. Which of these will win out in the months and years to come?

If ever a society seemed to be on the brink of something new it is Greece today, faced with a raw choice between death and rebirth. As this picture shows with fiery clarity, implacable forces are meeting immovable objects on the streets of Athens. The photograph captures a startling moment in riots a few days ago, as the Greek government tensely negotiated its latest bailout needs with the European Commission, the IMF and the European Central Bank. The photo makes it look as if demands for further sacrifices by the Greek people to an ideal of fiscal austerity are simply impossible. There have been many images of riot and revolution lately from all over the world, and a British interpreter of pictures of mayhem in Athens should be aware that our own cities offered some pretty juicy scenes of devastation last summer for the world to draw its conclusions from. Still, this is different.

The hate in this picture is so real. The eeriness of it lies in the intense light of firebombs at the heart of a group of police officers. This is not a dance between protesters and police with firebombs thrown theatrically: it looks more like attempted murder. There is a sense in this, and other images from Athens, of a conflict with no limits and the possibility of total war on the streets.

If democracy is Greek, so is violence. The ancient Greeks have sometimes, down the centuries, been pictured as a nation of philosophers, conversing calmly under lofty colonnades. This is the scene Raphael paints in his homage to Greek thought, The School of Athens, which depicts Plato, Aristotle and many more philosophers freely thinking and talking. But ancient Greeks were always ready to swap words for javelins. The first masterpiece of ancient Greek literature is Homer's Iliad, an epic of war. The separate, antagonistic city states of Greece found a sense of shared nationhood fighting the Persian empire. Soon they were fighting one another again. In his Republic, Plato offers some telling advice: Greeks, he says, when they fight, should not burn each others' cities and commit atrocities on fellow Greek civilians, as they are accustomed to do.

In ancient Greek terms, this picture is a battle of ideas. The protesters personify democracy. An ancient critic of democracy, such as Plato, might say they show the danger of democracy, that it degenerates into mob rule. Polybius, a Greek historian who wrote in the second century BC, drew on centuries of Greek political experience to argue that history is a cycle of political forms that decay and are replaced in turn – democracy gives way to demagogy, or to what later theorists gave the Greek name anarchy.

What do the police in the picture represent? They may not be the pure and disinterested elite of guardians that Plato advocates in his Republic as the ideal ruling class, but in this scene they are serving something not totally divorced from Plato. The Republic is the founding classic of political thought because it is the first book that ever reflected systematically on how people should govern one another. It is, however, a work of overwhelming idealism. We are like prisoners chained up in a cave, says Plato, trying to interpret shadows cast by torches on the cave wall. In other words, the messy perceptions we work with are utterly false and misleading, and the philosopher must learn to comprehend the realities beyond what is conventionally visible. The philosopher king will rule on the basis of ideal truth, not deceptive appearance.

This picture shows a philosophical ideal at war with visible reality. In this case the ideal is the eurozone, an abstract idea of a single currency imposed by economic guardians on messy realities that are now tearing and biting at it. Can this ideal monetary form survive? Or will the democratic crowd hurling firebombs bring the dream crashing down as so many human ideals have crashed over the centuries? Must Europe die where it began, on the bloody soil of Greece?

It is not fantastical to see in this picture a re-enactment of Plato's antithesis of democracy and an elite of experts and philosophers. Right now the leaders of the eurozone do seem to see democracy in Greece as a problem to be overcome, rather than a political treasure to be preserved. The euro ideal right now looks like… an ideal, in Plato's dangerous sense of a higher truth beyond the ken of the mob. What are those shadows in the firelight?